TurboTax only knows about your individual tax situation — not anybody else’s — so when the program says you are approved to claim your kids, you have met the conditions to claim your kids as dependents, as @fanfare says.
That doesn’t mean someone else may try to claim the same kids. However, if you entered your info correctly in TurboTax and the program said you are entitled to claim those kids as your dependents, the IRS will eventually rule in your favor if someone else tries to claim your kids.
As @xmasbaby0 says, if someone e-files a return before you do, claiming your kids, the IRS will reject your return. You will have to file by mail and face a long refund delay while the IRS sorts out who should claim the kids.
If you have questions, you can use use the IRS app that @xmasbaby0 mentions. You should also tell the person(s) who may try to claim your kids that they should not put those children on their return because you are claiming them.
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